The principal results of this study of remotely triggered seismicity in The Geysers geothermal field are the demonstration that triggering (initiation of earthquake failure) depends on a critical strain threshold and that the threshold level increases with decreasing frequency or equivalently, depends on strain rate. This threshold function derives from (1) analyses of dynamic strains associated with surface waves of the triggering earthquakes, (2) statistically measured aftershock zone dimensions, and (3) analytic functional representations of strains associated with power production and tides. The threshold is also consistent with triggering by static strain changes and implies that both static and dynamic strains may cause aftershocks. The observation that triggered seismicity probably occurs in addition to background activity also provides an important constraint on the triggering process. Assuming the physical processes underlying earthquake nucleation to be the same, Gomberg [this issue] discusses seismicity triggered by the MW 7.3 Landers earthquake, its constraints on the variability of triggering thresholds with site, and the implications of time delays between triggering and triggered earthquakes. Our results enable us to reject the hypothesis that dynamic strains simply nudge prestressed faults over a Coulomb failure threshold sooner than they would have otherwise. We interpret the rate-dependent triggering threshold as evidence of several competing processes with different time constants, the faster one(s) facilitating failure and the other(s) inhibiting it. Such competition is a common feature of theories of slip instability. All these results, not surprisingly, imply that to understand earthquake triggering one must consider not only simple failure criteria requiring exceedence of some constant threshold but also the requirements for generating instabilities.